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    "# Linear Regression Implementation from Scratch\n",
    "\n",
    "We will implement the entire method from scratch,\n",
    "including the data pipeline, the model,\n",
    "the loss function, and the minibatch stochastic gradient descent optimizer"
   ]
  },
  {
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    "origin_pos": 2,
    "tab": [
     "pytorch"
    ]
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   "source": [
    "%matplotlib inline\n",
    "import random\n",
    "import torch\n",
    "from d2l import torch as d2l"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
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   "source": [
    "Construct an artificial dataset\n",
    "according to a linear model with additive noise.\n",
    "The true parameters generating our dataset will be\n",
    "$\\mathbf{w} = [2, -3.4]^\\top$ and $b = 4.2$,\n",
    "and\n",
    "$$\\mathbf{y}= \\mathbf{X} \\mathbf{w} + b + \\mathbf\\epsilon$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "origin_pos": 7,
    "tab": [
     "pytorch"
    ]
   },
   "outputs": [],
   "source": [
    "def synthetic_data(w, b, num_examples):  \n",
    "    \"\"\"Generate y = Xw + b + noise.\"\"\"\n",
    "    X = torch.normal(0, 1, (num_examples, len(w)))\n",
    "    y = torch.matmul(X, w) + b\n",
    "    y += torch.normal(0, 0.01, y.shape)\n",
    "    return X, y.reshape((-1, 1))\n",
    "\n",
    "true_w = torch.tensor([2, -3.4])\n",
    "true_b = 4.2\n",
    "features, labels = synthetic_data(true_w, true_b, 1000)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Each row in `features` consists of a 2-dimensional data example\n",
    "and that each row in `labels` consists of a 1-dimensional label value (a scalar)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "origin_pos": 9,
    "tab": [
     "pytorch"
    ]
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "features: tensor([1.9054, 0.5880]) \n",
      "label: tensor([6.0026])\n"
     ]
    }
   ],
   "source": [
    "print('features:', features[0], '\\nlabel:', labels[0])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "origin_pos": 11,
    "tab": [
     "pytorch"
    ]
   },
   "outputs": [
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    "Define the `data_iter` function\n",
    "that\n",
    "takes a batch size, a matrix of features,\n",
    "and a vector of labels, yielding minibatches of the size `batch_size`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "origin_pos": 16,
    "tab": [
     "pytorch"
    ]
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "tensor([[-0.2553,  1.6106],\n",
      "        [-0.6488,  0.8568],\n",
      "        [ 0.4358,  1.3105],\n",
      "        [ 1.3671, -0.5341],\n",
      "        [ 0.1415, -0.5798],\n",
      "        [-0.4518, -0.5006],\n",
      "        [-1.8283, -1.0612],\n",
      "        [ 0.4615,  0.0652],\n",
      "        [ 0.1039, -0.6234],\n",
      "        [ 1.0851,  0.9522]]) \n",
      " tensor([[-1.7748],\n",
      "        [-0.0170],\n",
      "        [ 0.6168],\n",
      "        [ 8.7331],\n",
      "        [ 6.4507],\n",
      "        [ 4.9903],\n",
      "        [ 4.1400],\n",
      "        [ 4.9138],\n",
      "        [ 6.5235],\n",
      "        [ 3.1264]])\n"
     ]
    }
   ],
   "source": [
    "def data_iter(batch_size, features, labels):\n",
    "    num_examples = len(features)\n",
    "    indices = list(range(num_examples))\n",
    "    random.shuffle(indices)\n",
    "    for i in range(0, num_examples, batch_size):\n",
    "        batch_indices = torch.tensor(indices[i:min(i +\n",
    "                                                   batch_size, num_examples)])\n",
    "        yield features[batch_indices], labels[batch_indices]\n",
    "\n",
    "batch_size = 10\n",
    "\n",
    "for X, y in data_iter(batch_size, features, labels):\n",
    "    print(X, '\\n', y)\n",
    "    break"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Before we can begin optimizing our model's parameters\n",
    "we need to have some parameters in the first place"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "origin_pos": 19,
    "tab": [
     "pytorch"
    ]
   },
   "outputs": [],
   "source": [
    "w = torch.normal(0, 0.01, size=(2, 1), requires_grad=True)\n",
    "b = torch.zeros(1, requires_grad=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Define our model,\n",
    "relating its inputs and parameters to its outputs"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "origin_pos": 22,
    "tab": [
     "pytorch"
    ]
   },
   "outputs": [],
   "source": [
    "def linreg(X, w, b):  \n",
    "    \"\"\"The linear regression model.\"\"\"\n",
    "    return torch.matmul(X, w) + b"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Updating our model requires taking\n",
    "the gradient of our loss function,\n",
    "define the loss function first"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "origin_pos": 24,
    "tab": [
     "pytorch"
    ]
   },
   "outputs": [],
   "source": [
    "def squared_loss(y_hat, y):  \n",
    "    \"\"\"Squared loss.\"\"\"\n",
    "    return (y_hat - y.reshape(y_hat.shape))**2 / 2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Despite linear regression has a closed-form solution, other models in this book don't. Here we introduce minibatch stochastic gradient descent"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "origin_pos": 27,
    "tab": [
     "pytorch"
    ]
   },
   "outputs": [],
   "source": [
    "def sgd(params, lr, batch_size):  \n",
    "    \"\"\"Minibatch stochastic gradient descent.\"\"\"\n",
    "    with torch.no_grad():\n",
    "        for param in params:\n",
    "            param -= lr * param.grad / batch_size\n",
    "            param.grad.zero_()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Implement the main training loop"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "origin_pos": 32,
    "tab": [
     "pytorch"
    ]
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "epoch 1, loss 0.031524\n",
      "epoch 2, loss 0.000115\n",
      "epoch 3, loss 0.000049\n"
     ]
    }
   ],
   "source": [
    "lr = 0.03\n",
    "num_epochs = 3\n",
    "net = linreg\n",
    "loss = squared_loss\n",
    "\n",
    "for epoch in range(num_epochs):\n",
    "    for X, y in data_iter(batch_size, features, labels):\n",
    "        l = loss(net(X, w, b), y)\n",
    "        l.sum().backward()\n",
    "        sgd([w, b], lr, batch_size)\n",
    "    with torch.no_grad():\n",
    "        train_l = loss(net(features, w, b), labels)\n",
    "        print(f'epoch {epoch + 1}, loss {float(train_l.mean()):f}')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Evaluate our success in training\n",
    "by comparing the true parameters\n",
    "with those that we learned"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "origin_pos": 35,
    "tab": [
     "pytorch"
    ]
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "error in estimating w: tensor([ 0.0003, -0.0005], grad_fn=<SubBackward0>)\n",
      "error in estimating b: tensor([0.0002], grad_fn=<RsubBackward1>)\n"
     ]
    }
   ],
   "source": [
    "print(f'error in estimating w: {true_w - w.reshape(true_w.shape)}')\n",
    "print(f'error in estimating b: {true_b - b}')"
   ]
  }
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